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{\LARGE Mixed Distributions }
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Our text discusses random variables that are either discrete or continuous. We
will go further, and consider \emph{mixed} random variables that have a
distrete part and a continuous part. To justify this, consider observing a
lightbulb until it fails. What if there is a positive probability that the
failure time is zero (the bulb never goes on)?

Let $X_1$ be a discrete random variable, and let $X_2$ be (absolutely)
continuous.  You may think of a mixed random variable $Y$ as arising from a
two-step statistical experiment, like this. First, toss a coin with
probability of a Head equal to $\alpha$. If the coin shows Heads, $Y=X_1$; if
it is Tails, $Y=X_2$. Denoting by $C$ the outcome of the coin toss, we can
write the distribution function of Y as

\begin{eqnarray*} 
F_Y(y)  & = & P(Y \leq y) = P(Y \leq y|C=h)\,P(C=h) + P(Y \leq y|C=t)\,P(C=t) \\
        & = & \alpha P(X_1 \leq y) + (1-\alpha) P(X_2 \leq y) \\
        & = & \alpha F_{X_1}(y) + (1-\alpha) F_{X_2}(y)
\end{eqnarray*} 

Let $g(\cdot)$ be a function for which all the relevant expectations exist. By
the double expectation formula (which is actually part of the
\emph{definition} of conditional probability in more advanced courses), we
have

\begin{eqnarray*} 
E[g(Y)]  & = & E[E[g(Y)|C]] = [E[g(Y)|C=h]\,P(C=h) + [E[g(Y)|C=t]\,P(C=t) \\
        & = & \alpha E[g(X_1)] + (1-\alpha) E[g(X_2)] \\
        & = & \alpha \sum_x g(x)\,f_{X_1}(x) + 
            (1-\alpha) \int_{-\infty}^\infty g(x)\,f_{X_2}(x)\,dx
\end{eqnarray*} 

\enlargethispage*{1000 pt} This formula completely determines the distribution
of $Y$, since the function $g$ could be an indicator for any set of
interest. We can even use it to \emph{define} some notation that might
otherwise be confusing. Let us write

\begin{eqnarray*} 
E[g(Y)]     & = & \int g(y)\,dF_Y(y) = \int g(y)\,dP_Y(y) \\
            & = & \alpha \sum_x g(x)\,f_{X_1}(x) + 
                  (1-\alpha) \int_{-\infty}^\infty g(x)\,f_{X_2}(x)\,dx
\end{eqnarray*} 

You will prove in homework that this ``integral" enjoys all the usual
properties of sums and integrals. If you later learn that it is a special case
of a Lebesgue integral, no difficulty will arise. In the meantime, you will
have a concrete meaning for a notation that is frequently used without much
explanation.

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